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You only get charged for mobile data if you are actually connecting to the mobile data network. So of course using wi-fi doesn't count. he only time wi-fi would count is if one is using Verizon's mobile network as their main ISP and your using a mi-fi or jetpack. You're not. I have Charter as my ISP so why would Verizon charge me for data I use on Charter's network?
Anyway if yo want to make sure you are not using mobile data at all you can simple turn it off. just go to SETTINGS and where it says Mobile data just toggle it to off.